Calculating the complexity of an algorithm with 3 loops

Question

I tried to solve the following exercise :

What is the order of growth of the worst case running time of the following code fragment as a function of N?

int sum = 0;
for (int i = 1; i <= N; i++)
    for (int j = 1; j <= i*i; j++)
        for (int k = 1; k <= j*j; k++)
            sum++;

and I found that the complexity is O(n⁴), however the correct answer is :

The answer is : N⁷

For a given value of i, the body of the innermost loop is executed 1² + 2² + 3² + ... + (i²)² ~ 1/3 i⁶ times. Summing up over all values of i yields ~ 1/21 N⁷.

I would like some help to understand this answer and the correct way to calculate complexity in this case.

EDIT : In particular, I don't understand this statement :

1² + 2² + 3² + ... + (i²)² ~ 1/3 i⁶

Because for me, ² + 2² + 3² + ... + (i²)² ~ i⁴

Answer 1

EDIT:

I'll add a bit of explanation to clear up your confusion about the quote in your question. Let's consider a fixed value of i and focus on the innermost two loops:

for (int j = 1; j <= i*i; j++)
    for (int k = 1; k <= j*j; k++)
        sum++;

How many times is the j-loop iterated? The answer is i^2 times. On each of those iterations, the k-loop is iterated j^2 times, which is different for each outer iteration because the value of j increases from 1 all the way to i^2.

When j = 1, the k-loop iterates 1^2 times. When j = 2, the k-loop iterates 2^2 times. When j=3, 3^2 times. Tallying up the total number of the iterations of the k-loop over all values of j, you have 1^2 + 2^2 + 3^2 + ... + (i^2)^2, since j runs between 1 and i^2. Hopefully that clarifies how you arrive at the following statement:

For a given value of i, the body of the innermost loop is executed 1² + 2² + 3² + ... + (i²)² ~ 1/3 i⁶ times.

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The total number of iterations can be expressed in sum form. The innermost loop has exactly j^2 iterations for each (varying) value of j, the middle loop has i^2 iterations for each value of i, and the outermost loop has N iterations. More neatly, the exact number of iterations is:

Multiplying through, you'll find this is a 7th order polynomial in N, so it is apparent why this is O(N^7).

In case you doubt the answer above is correct, simply run your own code and compare the value of sum you get with the formula provided above:

var sum = 0;
var N = 10;

for (var i = 1; i <= N; i++)
    for (var j = 1; j <= i*i; j++)
        for (var k = 1; k <= j*j; k++)
            sum++;

function T(N) { return (1/420)*N*(1+N)*(1+2*N)*(8+11*N+21*N*N+20*N*N*N+10*N*N*N*N); }

console.log(sum===T(N));

Here's a demo: http://jsfiddle.net/wby9deax/. No matter what value of N you put in the answer will be correct (note: be careful with large values for N, it will probably freeze up your browser, since the number of iterations grows very rapidly).

Answer 2

int sum = 0;
for (int i = 1; i <= N; i++)            -- O(N^1)
    for (int j = 1; j <= i*i; j++)      -- O(N^2)
        for (int k = 1; k <= j*j; k++)  -- O(N^4)
            sum++;

Since they're nested (and since they're all linear) you get O(N¹ × N² × N⁴) = O(N¹⁺²⁺⁴) = O(N⁷)

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EDIT : In particular, I don't understand this statement :

1² + 2² + 3² + ... + (i²)² ~ 1/3 i⁶

Keep in mind that you may have N terms hiding in the "…" part.